where \(Ric\)  is the Ricci tensor on \(M\)  and Hess\(f\)  is the Hessian of \(f\) . We call (REF ) a generalized quasi–Einstein tensor. When \(\mu = \frac{1}{m}\)  for a positive integer \(m\) , the above is called a generalized \(m\) -quasi–Einstein manifold. We denote \(Ric_{f}^{\, m} := Ric + \mbox{\rm Hess}f - \frac{1}{m} \, df \otimes df\) , and we call \(Ric_{f}^{\, m}\)  a generalized \(m\) -quasi–Einstein tensor. The notion of generalized \(m\) -quasi–Einstein manifold was originated from the study of Einstein warped product manifolds (see [1]}). It plays an important role in the study of the weighted measure (c.f. [2]}). We also note that when \(m = \infty \) , a generalized \(m\) -quasi–Einstein tensor becomes
\(Ric_{f}^{\, \infty }:= Ric_{f} = Ric + \mbox{\rm Hess}f.\) 
